Multi-Parameter Persistent Homology for Poverty Trap Detection

Introduction

The concept of a poverty trap — a configuration in which low income is self-reinforcing and exit is structurally blocked — has strong theoretical foundations in economics and sociology. Yet operationalising the distinction between a poverty trap and extended transient poverty has proved analytically challenging. Standard persistence methods analyse a single filtration parameter; they cannot simultaneously condition on both the depth and the temporal extent of poverty spells.

Multi-parameter persistent homology (MPH) — a generalisation to two or more filtration parameters — provides exactly this capability. This paper applies MPH to income-augmented employment trajectory data, constructing a two-parameter persistence module that captures both income-level and temporal dimensions of poverty experience.

Background

Multi-Parameter Persistence

In single-parameter persistent homology, a filtration $\{X_t\}_{t \in \mathbb{R}}$ is indexed by a single real parameter. Multi-parameter persistence extends this to $\{X_{s,t}\}_{(s,t) \in \mathbb{R}^2}$, indexed by a pair of parameters. The resulting persistence module is more complex — indecomposable modules are generically wild — but invariants such as the rank function and fibered barcodes remain computationally tractable and statistically useful.

Poverty Traps in Sociological Literature

The sociological literature on poverty dynamics identifies persistent poverty as qualitatively distinct from recurrent poverty, but operationalisation is typically threshold-based (e.g., five or more years below 60% median income). MPH provides a geometry-respecting alternative.

Methods

Employment trajectories are augmented with annual equivalised household income from Understanding Society and BHPS. A two-parameter Rips filtration is constructed indexing simultaneously by income depth (normalised distance below poverty threshold) and time (calendar year of observation). The two-parameter persistence module is approximated via the fibered barcode method. GPU computation using the multipers 2.3.4 library is used to make the fibered barcode computation practical at scale (development-scale bifiltration: ~3s for 2,000 maxmin landmarks; full 8,000-landmark production run with GPU acceleration planned).

Data

Understanding Society waves 1–14 provide the core data. BHPS waves 1–18 are used for historical comparison. Income is equivalised using the modified OECD scale.

Results

The Bifiltration: Computation and Discretisation

A development-scale Rips × income bifiltration (2,000 maxmin landmarks drawn from the 27,280-trajectory P01 checkpoint) produces 215,958 simplices in approximately 3 seconds. Income score is computed as the proportion-weighted mean of per-wave income bands (Low = 0, Mid = 1, High = 2) with mean 1.02 and median 1.00. Comparison of quantile-based and fixed-grid income discretisation strategies confirms that the quantile grid produces more balanced H₀ low-fraction (0.512 vs 0.454) and more stable H₁ convergence properties. Quantile grid is adopted as the primary analysis; fixed grid is used in the sensitivity appendix.

The Poverty Trap Signature: Absence of H₁ at Low Income

The central finding inverts the initial hypothesis. The poverty trap signature in the bifiltration is not the presence of isolated H₀ components (isolated connected components surviving across both income depth and time dimensions). It is the absence of H₁ at low income levels. H₁ signed-measure mass shows a strong income gradient: Q3/Q4 income quartiles carry approximately 4× the H₁ mass of Q1. Low-income trajectories form simpler, more linear topological structures — they lack the diverse inter-trajectory connectivity that characterises mid-to-high income trajectory space. The poverty trap is, topologically, a region of impoverished connectivity rather than isolated enclosure.

Topological Interpretation

This finding reframes the poverty trap concept geometrically. High-income trajectories are diverse: they connect in complex configurations because the routes to sustained high income are varied (sector changes, qualification effects, partnership effects). Low-income trajectories converge on a narrow structural form — a topologically simple manifold from which the paths out are few and the paths in are many. The bifiltration captures this as the relative absence of H₁ features in the low-income end of the income filtration. GPU-scale computation (full 8,000-landmark production run) is planned to confirm the gradient at production scale.

Discussion

The H₁ absence finding has direct implications for policy design. If poverty traps were characterised by isolated H₀ components, the intervention implication would be to build connections between isolated sub-populations and the main labour market. If they are characterised by H₁ absence — by structural simplicity rather than enclosure — the implication is that poverty trap exit requires diversifying the trajectory options available from low-income starting points, not bridging a gap between separate manifolds. MPH provides a geometry-respecting framework for distinguishing these two structural diagnoses that threshold-based operationalisations of poverty traps cannot.

Conclusion

Exploratory bifiltration results indicate that poverty trap topology in UK panel data is characterised by the absence of H₁ at low income levels rather than by isolated H₀ components, inverting the initial hypothesis and reframing the geometry of structural disadvantage. Full-scale GPU computation is underway to confirm the gradient and enable spatial disaggregation by NUTS-2 region.